Tamagawa Number Conjecture for Zeta Values 165

Home , Euler system, Selmer group

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(often conjectural) say that the order of zero or pole of the zeta function at an integer point is equal to the rank of the related finitely generated arithmetic abelian group (Tate, the conjecture (0.3), Beilinson, Bloch, ...) and the value of the zeta function at an integer point is related to the order of the related arithmetic finite group. In [BK], Bloch and the author formulated a general conjecture on (0.0) (Tama- gawa number conjecture for motives). Further generalizations of Tamagawa number conjecture by Fontaine, Perrin-Riou, and the author [FP], [Pe1] [Ka1], [Ka2] have the form (0.4) zeta functions (= Euler products, analytic) ↔ zeta elements (= Euler systems, arithmetic) ↔ arithmetic groups. Here the first ↔ means that zeta functions enter the arithmetic world transforming themselves into zeta elements, and the second ↔ means that zeta elements generate “determinants” of certain etale cohomology groups. The aim of this paper is to discuss (0.4) in an expository style. We review (0.1) in §1, and then in §2, we describe the generalized Tamagawa number conjecture (0.4), the relation with (0.2), and an application of the philosophy (0.4) to (0.3). In this paper, we fix a prime number p. For a commutative ring R, let Q(R) be the total quotient ring of R obtained from R by inverting all non-zerodivisors. 1. Grothendieck formula and zeta elements

Let X be a scheme of finite type over a finite field Fq. We assume p is different from char(Fq). In this §1, we first review the formula (1.1.2) of Grothendieck representing zeta functions of p-adic sheaves on X by etale cohomology. We then show that those zeta functions are recovered from p-adic zeta elements (1.3.5). 1.1. Zeta functions and etale cohomology groups in positive characteristic case. −s −1 The Hasse zeta function ζ(X,s) = Qx∈|X| (1 − ♯κ(x) ) , where |X| denotes the set of all closed points of x and κ(x) denotes the residue field of x, −s has the form ζ(X,s)= ζ(X/Fq, q ) where

deg(x) −1 ζ(X/Fq,u)= Y (1 − u ) , deg(x) = [κ(x): Fq]. (1.1.1) x∈|X|

A part of Weil conjectures was that ζ(X/Fq,u) is a rational function in u, and it was proved by Dwork and then slighly later by Grothendieck. The proof of Grothendieck gives a presentation of ζ(X/Fq,u) by using etale cohomologyy. More generally, for a ﬁnite extension L of Qp and for a constructible L-sheaf F on X, Grothendieck proved that the L-function L(X/Fq, F,u) has the presentation

−1 m ¯ (−1)m L(X/Fq, F,u)= Y detL(1 − ϕqu ; Het,c(X ⊗Fq Fq, F)) (1.1.2) m Tamagawa Number Conjecture for zeta Values 165

m where Het,c is the etale cohomology with compact supports and ϕq is the action of the q-th power morphism on X. In the case L = Qp = F, ζ(X/Fq,u)= L(X/Fq, F,u). 1.2. p-adic zeta elements in positive characteristic case. Determinants appear in the theory of zeta functions as above, rather often. The regulator of a number field, which appears in the class number formula, is a determinant. Such relation with determinant is well expressed by the notion of “determinant module”. If R is a field, for an R-module V of dimension r, detR(V ) means the 1 dimen- r sional R-module ∧R(V ). For a bounded complex C of R-modules whose cohomolo- m m ⊗(−1)m gies H (C) are finite dimensional, detR(C) means ⊗m∈Z {detR(H (C))} . This definition is generalized to the definition of an invertible R-module detR(C) associated to a perfect complex C of R-modules for a commutative ring R (see −1 [KM]). detR (C) means the inverse of the invertible module detR(C). By a pro-p ring, we mean a topological ring which is an inverse limit of finite rings whose orders are powers of p. Let Λ be a commutative pro-p ring. By a ctf Λ-complex on X, we mean a complex of Λ-sheaves on X for the etale topology with constructible cohomology sheaves and with perfect stalks. For a ctf Λ-complex F on X, RΓet,c(X, F) (c means with compact supports) is a perfect complex over Λ. For a commutative pro-p ring Λ and for a ctf Λ-complex F on X, we define the −1 p-adic zeta element ζ(X, F, Λ) which is a Λ-basis of detΛ RΓet,c(X, F). Consider the distinguished triangle

¯ ¯ RΓet,c(X, F) → RΓet,c(X ⊗Fq Fq, F)@ > 1 − ϕ>>RΓet,c(X ⊗Fq Fq, F). (1.2.1)

Since det is multiplicative for distinguished triangles, (1.2.1) induces an isomorphism

−1 −1 ∼ F ¯ F ¯ ∼ detΛ RΓet,c(X, F) = detΛ RΓet,c(X ⊗ q Fq, F) ⊗Λ detΛRΓet,c(X ⊗ q Fq, F) = Λ. (1.2.2) −1 We deﬁne ζ(X, F, Λ) to be the image of 1 ∈ Λ in detΛ RΓet,c(X, F) under (1.2.2). −1 It is a Λ-basis of the invertible Λ-module detΛ RΓet,c(X, F). 1.3. Zeta functions and p-adic zeta elements in positive characteristic case. Let L be a ﬁnite extension of Qp, let OL be the valuation ring of L, and let F be a constructible OL-sheaf on X. We show that the zeta function

L(X/Fq, FL,u) of the L-sheaf FL = F ⊗OL L is recovered from a certain p-adic zeta element as in (1.3.5) below. Let

¯ Λ= OL[[Gal(Fq/Fq)]] = lim←− OL[Gal(Fqn /Fq)]. (1.3.1) n

Let s(Λ) be the Λ-module Λ which is regarded as a sheaf on the etale site of X via the natural action of Gal(F¯ q/Fq). Then

m m ∼ F F n Het,c(X, F ⊗OL s(Λ)) = ←−lim Het,c(X ⊗ q q , F) (1.3.2) n 166 KazuyaKato

where the transition maps of the inverse system are the trace maps. From this, we m can deduce that Het,c(X, F⊗OL s(Λ)) is a ﬁnitely generated OL-module for any m.

Hence we have Q(Λ)⊗Λ RΓet,c(X, F⊗OL s(Λ)) = 0 and this gives an identiﬁcatition canonical isomorphism

−1 Q(Λ) ⊗Λ detΛ RΓet,c(X, F ⊗OL t(Λ)) = Q(Λ). (1.3.3)

Note n Q(Λ) = Q(lim←− OL[u]/(u − 1)) ⊃ Q(OL[u]) = L(u). (1.3.4) n By a formal argument, we can prove the following (1.3.5) (1.3.6) which show zeta function = zeta element, zeta value = zeta element, respectively.

L(X/Fq, FL,u)= ζ(X, F ⊗OL s(Λ), Λ) in Q(Λ). (1.3.5)

m If Het,c(X, FL) =0 for any m, L(X/Fq, FL,u) has no zero or pole at u = 1, and

L(X/Fq, FL, 1) = ζ(X, F, OL) in L. (1.3.6) 2. Tamagawa number conjecture In 2.1, we describe the generalized version of Tamagawa number conjecture. In 2.2 (resp. 2.3), we consider p-adic zeta elements associated to 1 (resp. 2) dimensional p-adic representations of Gal(Q¯ /Q), and their relations to (0.2) (resp. (0.3)). 1 2.1. The conjecture. Let X be a scheme of finite type over Z[ p ]. For a complex of sheaves F on X for the etale topology, we define the compact support version RΓet,c(X, F) of RΓet(X, F) as the mapping fiber of

1 RΓet(Z[ ],Rf F) → RΓet(R,Rf F) ⊕ RΓet(Qp,Rf F). p ! ! !

1 where f : X → Spec(Z[ p ]). It can be shown that for a commutative pro-p ring Λ and for a ctf Λ-complex F on X, RΓet,c(X, F) is perfect. The following is a generalized version of the Tamagawa number conjecture [BK] (see [FP], [Pe1], [Ka1], [Ka2]). In [BK], the idea of Tamagawa number of motives was important, but it does not appear explicitly in this version. Conjecture. To any triple (X, Λ, F) consisting of a scheme X of ﬁnite type over 1 Z[ p ], a commutative pro-p ring Λ, and a ctf Λ-complex on X, we can associate a Λ-basis ζ(X, F, Λ) of

−1 ∆(X, F, Λ) = detΛ RΓet,c(X, F), Tamagawa Number Conjecture for zeta Values 167

which we call the p-adic zeta element associated to F, satisfying the following conditions (2.1.1)-(2.1.5). (2.1.1) If X is a scheme over a finite field Fq, ζ(X, F, Λ) coincides with the element defined in §3.2. (2.1.2) (rough form) If F is the p-adic realization of a motive M, ζ(X, F, Λ) −e recovers the complex value lims→0 s L(M,s) where L(M,s) is the zeta function of M and e is the order of L(M,s) at s =0. ′ ′ L (2.1.3) If Λ is a pro-p ring and Λ → Λ is a continuous homomorphism, ζ(X, F⊗Λ ′ ′ L ′ ′ ∼ Λ , Λ ) coincides with the image of ζ(X, F, Λ) under ∆(X, F⊗ΛΛ , Λ ) = ∆(X, F)⊗Λ Λ′. (2.1.4) For a distinguished triangle F ′ →F→F” with common X and Λ, we have

′ ′ ζ(X, F, Λ) = ζ(X, F , Λ)⊗ζ(X, F”, Λ) in ∆(X, F, Λ) = ∆(X, F , Λ)⊗Λ∆(X, F”, Λ).

1 (2.1.5) If Y is a scheme of ﬁnite type over Z[ p ] and f : X → Y is a separated morphism,

ζ(Y,Rf!F, Λ) = ζ(X, F, Λ) in ∆(Y,Rf!F, Λ) = ∆(X, F, Λ).

By this (2.1.5), the constructions of p-adic zeta elements are reduced to the 1 case X = Spec(Z[ p ]). How to formulate the part (4.1.2) of this conjecture is reduced to the case of motives over Q by (2.1.5) and L(M,s)= L(Rf!(M),s) (by philosophy 1 of motives), where f : X → Spec(Z[ p ]). The conditions (2.1.3)-(2.1.5) are formal properties which are analogous to formal properties of zeta functions. The conditions (2.1.1) and (2.1.3)-(2.1.5) can be interpreted as (2.1.6) The system (X, Λ, F) 7→ ζ(X, F, Λ) is an “Euler system”. In fact, let L be a finite extension of Qp, S a finite set of prime numbers containing p, and let T be a free OL-module of finite rank endowed with a continuous OL-linear action of Gal(Q¯ /Q) which is unramified outside S. For m ≥ 1, let Rm = OL[Gal(Q(ζm)/Q)] and let Z 1 −1 Z 1 zm = ζRm ( [ p ], jm,!(T ⊗OL s(Rm)), Rm) ∈ detRm RΓet,c( [ζm, mS ],T ). 1 1 (jm : Spec(Z[ mS ]) → Spec(Z[ p ])). Then the conditions (4.1.1) and (4.1.3)-(4.1.5) tell that when m varies, the p-adic zeta elements zm form a system satisfying the conditions of Euler systems formulated by Kolyvagin [Ko]. We illustrate the relation (2.1.2) with zeta functions. Let M be a motive over Q, that is, a direct summand of the motive Hm(X)(r) for a proper smooth scheme X over Q and for r ∈ Z, and assume that M is endowed with an action of a number field K. Then the zeta function L(M,s) lives in C, and the p-adic zeta element lives in the world of p-adic etale cohomology. Since these two worlds are too much different in nature, L(M,s) and the p-adic zeta element are not simply related. 168 KazuyaKato

However in the middle of C and the p-adic world, (a) there is a 1 dimensional K-vector space ∆K (M) constructed by the Betti realization and the de Rham realization of M, and K-groups (or motivic cohomology groups) associated to M. Let ∞ be an Archimedean place of K. Then (b) there is an isomorphism

∆K (M) ⊗K K∞@ >>=∼>K∞ constructed by Hodge theory and K-theory. Let w be a place of K lying over p, let Mw be the representation of Gal(Q¯ /Q) ¯ over Kw associated to M, and let T be a Gal(Q/Q)-stable OKw -lattice in Mw. Then (c) there is an isomorphism

−1 1 ∆K (M) ⊗K Kw@ >>=∼> det RΓet,c(Z[ ], j∗Mw) Kw p

−1 1 =det RΓet,c(Z[ ], j∗T ) ⊗O Kw OKw p Kw

1 where j : Spec(Q) → Spec(Z[ p ]), constructed by p-adic Hodge theory and K-theory. See [FP] how to construct (a)-(c) (constructions require some conjectures). The part (2.1.2) of the conjecture is: (d) there exists a K-basis ζ(M) of ∆K (M) (called the rational zeta ele- −e ment associated to M), which is sent to lims→0 s L(M,s) under the isomor- 1 phism (b) where e is the order of L(M,s) at s = 0, and to ζ(Z[ p ], j∗T, OKw ) −1 Z 1 in detKw RΓet,c( [ p ], j∗Mw) under the isomorphism (c). −e The existence of ζ(M) having the relation with lims→0 s L(M,s) was conjectured by Beilinson [Be]. How zeta functions and p-adic zeta elements are related is illustrated in the following diagram. zeta functions side@.(Betti)@ < Hodge theory << (de Rham)@. @.@AregulatorAA@AAp-adic Hodge theoryA@.

@.(K-theory)@ >> Chern class > (etale)@.p-adic zeta elements side. We have the following picture. automorphic rep@ >>p-adic Gal rep @V V V @V V ?V @V V ?V zeta [email protected] zeta [email protected] zeta elememts The left upper arrow with a question mark shows the conjecture that the map {motives}→{zeta functions} factor through automorphic representations, which Tamagawa Number Conjecture for zeta Values 169

is a subject of non-abelian class field theory (Langlands correspondences). As the other question marks indicate, we do not know how to construct zeta elements in general, at present. 2.2. p-adic zeta elements for 1 dimensional galois representations. Let Λ be a commutative pro-p ring, and assume we are given a continuous homomorphism ρ : Gal(Q¯ /Q) → GLn(Λ) which is unramified outside a finite set S of prime numbers S containing p. Let ⊕n ¯ 1 F =Λ on which Gal(Q/Q) acts via ρ, regarded as a sheaf on Spec(Z[ S ]) for the 1 etale topology. We consider how to construct the p-adic zeta element ζ(Z[ S ], F, Λ). In the case n = 1, we can use the “universal objects” as follows. Such ρ comes from the canonical homomorphism

ρuniv : Gal(Q¯ /Q) → GL1(Λuniv) where Λuniv = Zp[[Gal(Q(ζNp∞ )/Q)]] for some N ≥ 1 whose set of prime divisors coincide with S and for some continuous ∼ 1 ring homomorphism Λuniv → Λ. We have F = Funiv ⊗Λuniv Λ. Hence ζ(Z[ S ], F, Λ) 1 should be deﬁned to be the image of ζ(Z[ S ], Funiv, Λuniv). As is explained in [Ka2] 1 Ch. I, 3.3, ζ(Z[ S ], Funiv, Λuniv) is the pair of the p-adic Riemann zeta function and a system of cyclotomic units. Iwasawa main conjecture is regarded as the statemnet 1 that this pair is a Λuniv-basis of ∆(Z[ S ], Funiv, Λuniv). 2.3. p-adic zeta elements for 2 dimensional Galois representations. Now consider the case n = 2. The works of Hida, Wiles, and other people suggest that the universal objects Λuniv and Funiv for 2 dimensional Galois representations in which the determinant of the action of the complex conjugation is -1, are given by n Λuniv = lim←− p-adic Hecke algebras of weight 2 and of level Np , n 1 n Funiv = lim←− H of modular curves of level Np . n

Beilinson [Be] discovered ratinal zeta elements in K2 of modular curves, and the im- ages of these elements in the etale cohomology under the Chern class maps become p-adic zeta elements, and the inverse limit of these p-adic zeta elements should be 1 ζ(Z[ S ], Funiv, Λuniv) at least conjecturally. By using this plan, the author obtained p-adic zeta elements for motives associated to eigen cusp forms of weight ≥ 2, from Beilinson elements. Here it is not yet proved that these p-adic zeta elements are actually basis of ∆, but it can be proved that they have the desired relations with values L(E, χ, 1) and L(f,χ,r) (1 ≤ r ≤ k − 1) for elliptic curves over Q (which are modular by [Wi], [BCDT]) and for eigen cusp forms of weight k ≥ 2, and for Dirichlet charcaters χ. Beilinson elements are related in the Archimedean world −1 to lims→0 s L(E,χ,s) for elliptic curves E over Q, but not related to L(E, χ, 1). However since they become universal (at least conjecturally) in the inverse limit in 170 KazuyaKato the p-adic world, we can obtain from them p-adic zeta elements related to L(E, χ, 1). Using these elements and applying the method of Euler systems [Ko], [Pe2], [Ru2], [Ka3], we can obtain the following results ([Ka4]).

Theorem. Let E be an elliptic curve over Q, let N ≥ 1, and let χ : Gal(Q(ζN )/Q) × =∼ (Z/NZ) → C be a homomorphism. If L(E, χ, 1) 6= 0, the χ-part of E(Q(ζN )) and the χ part of the Tate-shafarevich group of E over Q(ζN ) are ﬁnite.

The p-adic L-function Lp(E) of E is constructed from the values L(E, χ, 1). Theorem. Let E be an elliptic curve over Q which is of good reduction at p. (1) rank(E(Q) ≤ ords=1Lp(E). (2) Assume E is ordinary at p. Let Λ = Zp[[Gal(Q(ζp∞ /Q)]]. Then the p-primary Selmer group of E over Q(ζp∞ ) is Λ-cotorsion and its characteristic n polynomial divides p Lp(E) for some n. This result was proved by Rubin in the case of elliptic curves with complex multiplication ([Ru1]). As described above, we can obtain p-adic zeta elements of motives associated to eigen cusp forms of weight ≥ 2. For such modular forms, we can prove the analogous statement as the above (2). Mazur and Greenberg conjectured that the charcteristic polynomial of the above p-primary Selmer group and the p-adic L-function divide each other. References [Be] Beilinson, A., Higher regulators and values of L-functions, J. Soviet Math., 30 (1985), 2036–2070. [BK] Bloch, S. and Kato, K., Tamagawa numbers of motives and L-functions, in The Grothendieck Festschrift, 1, Progress in Math., 86, Burkhauser (1990), 333–400. [BCDT] Breuil, C., Conrad, B, Diamond, F., Taylor, R., On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc., 14 (2001), 834–939. [FP] Fontaine, J. -M., and Perrin-Riou, B., Autour des conjectures de Bloch et Kato, cohomologie Galoisienne et valeurs de fonctions L, Proc. Symp. Pure Math. 55, Amer. Math. Soc., (1994), 599–706. [Ka1] Kato, K., Iwasawa theory and p-adic Hodge theory, Kodai Math. J., 16 (1993), 1–31. [Ka2] Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. I, Arithmetic algebraic geometry (Trento, 1991), 50–163, Lecture Notes in Math., 1553, Springer, Berlin (1993). [Ka3] Kato, K., Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J., 22 (1999), 313–372. [Ka4] Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, preprint. Tamagawa Number Conjecture for zeta Values 171

[KM] Knudsen, F., and Mumford, D., The projectivity of the moduli space of stable curves I, Math. Scand., 39, 1 (1976), 19–55. [Ko] Kolyvagin, V. A., Euler systems, in The Grothendieck Festchrift, 2, Birk- houser (1990), 435–483. [Pe1] Perrin-Riou, B., Fonction L p-adiques des représentations p-adiques, Astérisque 229 (1995). [Pe2] Perrin-Riou, B., Systemes d’Euler p-adiques et théorie d’Iwaswa, Ann. Inst. Fourier, 48 (1998), 1231–1307. [Ru1] Rubin, K., The “main conjecture” of Iwasawa theory for imaginary qua- dratic fields, Inventiones math., 103 (1991), 25–68. [Ru2] Rubin, K., Euler systems, Hermann Weyl Lectures, Annals of Math. Stud- ies, 147, Princeton Univ. Press (2000). [Wi] Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math., 141 (1995), 443–551.